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Properties

Label 580.2.c.b
Level $580$
Weight $2$
Character orbit 580.c
Analytic conductor $4.631$
Analytic rank $0$
Dimension $10$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [580,2,Mod(349,580)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("580.349"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(580, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 580 = 2^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 580.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.63132331723\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 22x^{8} + 149x^{6} + 324x^{4} + 252x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{7} q^{5} + (\beta_{9} + \beta_{8} - \beta_1) q^{7} + (\beta_{6} + \beta_{5} + \beta_{4} + \cdots - 1) q^{9} + (\beta_{8} + \beta_{7} - \beta_{2}) q^{11} + ( - \beta_{9} - \beta_{7} + \cdots + \beta_{3}) q^{13}+ \cdots + (2 \beta_{8} + 2 \beta_{7} - \beta_{6} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 14 q^{9} - 6 q^{15} - 4 q^{19} + 24 q^{21} - 4 q^{25} - 10 q^{29} + 16 q^{31} - 24 q^{35} - 4 q^{39} + 28 q^{41} - 22 q^{45} - 26 q^{49} + 32 q^{51} - 26 q^{55} + 24 q^{59} + 52 q^{61} - 34 q^{65}+ \cdots + 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 22x^{8} + 149x^{6} + 324x^{4} + 252x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{8} + 22\nu^{6} + 147\nu^{4} + 294\nu^{2} + 144 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{9} + 2\nu^{8} - 21\nu^{7} + 43\nu^{6} - 128\nu^{5} + 275\nu^{4} - 196\nu^{3} + 490\nu^{2} - 56\nu + 208 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{8} + 64\nu^{6} + 403\nu^{4} + 690\nu^{2} + 288 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{9} - 4\nu^{8} - 64\nu^{7} - 86\nu^{6} - 405\nu^{5} - 550\nu^{4} - 708\nu^{3} - 980\nu^{2} - 284\nu - 416 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3\nu^{9} - 4\nu^{8} + 64\nu^{7} - 86\nu^{6} + 405\nu^{5} - 550\nu^{4} + 708\nu^{3} - 980\nu^{2} + 284\nu - 416 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -3\nu^{9} - 6\nu^{8} - 64\nu^{7} - 128\nu^{6} - 405\nu^{5} - 810\nu^{4} - 712\nu^{3} - 1420\nu^{2} - 316\nu - 600 ) / 8 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3\nu^{9} - 6\nu^{8} + 64\nu^{7} - 128\nu^{6} + 405\nu^{5} - 810\nu^{4} + 712\nu^{3} - 1420\nu^{2} + 316\nu - 600 ) / 8 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 8\nu^{9} + 3\nu^{8} + 171\nu^{7} + 64\nu^{6} + 1085\nu^{5} + 405\nu^{4} + 1912\nu^{3} + 710\nu^{2} + 816\nu + 300 ) / 4 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{5} + \beta_{4} + \beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} - 8\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{8} - \beta_{7} - 10\beta_{6} - 10\beta_{5} - 12\beta_{4} - 10\beta_{2} + 34 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{9} - 9\beta_{8} + 7\beta_{7} + 15\beta_{6} - 13\beta_{5} + 2\beta_{3} + 74\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 19\beta_{8} + 19\beta_{7} + 94\beta_{6} + 94\beta_{5} + 130\beta_{4} + 100\beta_{2} - 334 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 42\beta_{9} + 69\beta_{8} - 27\beta_{7} - 185\beta_{6} + 155\beta_{5} - 30\beta_{3} - 710\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -271\beta_{8} - 271\beta_{7} - 892\beta_{6} - 892\beta_{5} - 1390\beta_{4} - 1020\beta_{2} + 3382 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -626\beta_{9} - 493\beta_{8} - 133\beta_{7} + 2159\beta_{6} - 1789\beta_{5} + 370\beta_{3} + 6950\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/580\mathbb{Z}\right)^\times\).

\(n\) \(117\) \(291\) \(321\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
349.1
− 3.27452i
− 2.85932i
− 1.35708i
− 0.809011i
− 0.778250i
0.778250i
0.809011i
1.35708i
2.85932i
3.27452i
0 − 3.27452i 0 1.98729 − 1.02503i 0 1.44995i 0 −7.72245 0
349.2 0 − 2.85932i 0 −1.22199 + 1.87263i 0 3.01459i 0 −5.17570 0
349.3 0 − 1.35708i 0 0.202675 − 2.22686i 0 − 0.415357i 0 1.15832 0
349.4 0 − 0.809011i 0 −2.14950 − 0.616164i 0 3.65075i 0 2.34550 0
349.5 0 − 0.778250i 0 1.18153 − 1.89842i 0 − 4.82798i 0 2.39433 0
349.6 0 0.778250i 0 1.18153 + 1.89842i 0 4.82798i 0 2.39433 0
349.7 0 0.809011i 0 −2.14950 + 0.616164i 0 − 3.65075i 0 2.34550 0
349.8 0 1.35708i 0 0.202675 + 2.22686i 0 0.415357i 0 1.15832 0
349.9 0 2.85932i 0 −1.22199 − 1.87263i 0 − 3.01459i 0 −5.17570 0
349.10 0 3.27452i 0 1.98729 + 1.02503i 0 − 1.44995i 0 −7.72245 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 349.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 580.2.c.b ✓ 10
3.b odd 2 1 5220.2.g.d 10
4.b odd 2 1 2320.2.d.i 10
5.b even 2 1 inner 580.2.c.b ✓ 10
5.c odd 4 2 2900.2.a.m 10
15.d odd 2 1 5220.2.g.d 10
20.d odd 2 1 2320.2.d.i 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
580.2.c.b ✓ 10 1.a even 1 1 trivial
580.2.c.b ✓ 10 5.b even 2 1 inner
2320.2.d.i 10 4.b odd 2 1
2320.2.d.i 10 20.d odd 2 1
2900.2.a.m 10 5.c odd 4 2
5220.2.g.d 10 3.b odd 2 1
5220.2.g.d 10 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} + 22T_{3}^{8} + 149T_{3}^{6} + 324T_{3}^{4} + 252T_{3}^{2} + 64 \) acting on \(S_{2}^{\mathrm{new}}(580, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + 22 T^{8} + \cdots + 64 \) Copy content Toggle raw display
$5$ \( T^{10} + 2 T^{8} + \cdots + 3125 \) Copy content Toggle raw display
$7$ \( T^{10} + 48 T^{8} + \cdots + 1024 \) Copy content Toggle raw display
$11$ \( (T^{5} - 39 T^{3} + \cdots + 44)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + 110 T^{8} + \cdots + 313600 \) Copy content Toggle raw display
$17$ \( T^{10} + 140 T^{8} + \cdots + 2768896 \) Copy content Toggle raw display
$19$ \( (T^{5} + 2 T^{4} - 38 T^{3} + \cdots + 80)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + 80 T^{8} + \cdots + 30976 \) Copy content Toggle raw display
$29$ \( (T + 1)^{10} \) Copy content Toggle raw display
$31$ \( (T^{5} - 8 T^{4} + \cdots - 692)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + 164 T^{8} + \cdots + 1048576 \) Copy content Toggle raw display
$41$ \( (T^{5} - 14 T^{4} + \cdots - 16000)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + 166 T^{8} + \cdots + 7573504 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 128686336 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 1436713216 \) Copy content Toggle raw display
$59$ \( (T^{5} - 12 T^{4} + \cdots - 10048)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} - 26 T^{4} + \cdots + 27584)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + 408 T^{8} + \cdots + 57032704 \) Copy content Toggle raw display
$71$ \( (T^{5} - 12 T^{4} + \cdots - 8384)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + 300 T^{8} + \cdots + 262144 \) Copy content Toggle raw display
$79$ \( (T^{5} + 4 T^{4} + \cdots + 10892)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + 312 T^{8} + \cdots + 4000000 \) Copy content Toggle raw display
$89$ \( (T^{5} + 10 T^{4} + \cdots + 112448)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + 476 T^{8} + \cdots + 76877824 \) Copy content Toggle raw display
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